The two equations that are not laws of motion

Maxwell’s four equations are usually written in a block, as though they were four statements of the same kind. Look at what each one differentiates and the block splits in two.

Faraday’s law and Ampère’s law with Maxwell’s term each say how a field changes. They are equations of motion, and given a field now they give the field a moment later.

Gauss’s law and the statement that there are no magnetic charges contain no time derivative at all. They are conditions on the field at one instant, and there is nothing in them to integrate.

What the figure actually does #

The point can be made by construction rather than by argument. Put a field on a grid, integrate the two curl equations, and never look at the other two.

The residuals of the two constraints are plotted, and they sit at $2.4\times10^{-16}$ of the field’s own size for the whole run — machine rounding — while the field propagates and its amplitude changes by a factor of $8.6$.

That is not a coincidence of the numerical scheme, and it is not a small violation that happens to stay small. It is exact, and the reason is an identity — the same one that lets a field be written as the curl of a potential in the first place.

Take the divergence of Faraday’s law. The right-hand side becomes the divergence of a curl, which vanishes for any vector field whatever, so

$$\frac{\partial}{\partial t}(\nabla\cdot\mathbf{B}) = 0.$$

Whatever the fields are doing, the divergence of $\mathbf{B}$ cannot change. Set it to zero once and it is zero for ever.

The same manoeuvre on Ampère’s law gives

$$\frac{\partial}{\partial t}(\nabla\cdot\mathbf{D}) = -\nabla\cdot\mathbf{J},$$

which is not identically zero. It is zero exactly when the charge density is obeying the continuity equation.

A last observation about what the figure does not show. The residual sits at rounding for the whole run, which demonstrates that the constraints are preserved and says nothing about whether they were true to begin with. They were, because the initial field was built to satisfy them — two Gaussian pulses in components whose divergences vanish identically. Had the initial field violated them, the violation would have sat there unchanged for the whole run, propagating as a spurious static charge and a spurious magnetic charge that nothing could remove. That is the practical form the statement takes: an error in the initial data is permanent, and an error made during the evolution is impossible.

Two constraints, one condition #

So the two are not on the same footing after all.

The magnetic one is unconditional. Nothing in the theory can change $\nabla\cdot\mathbf{B}$, so there is no process by which a magnetic charge could appear. That is a much stronger statement than “none has been found.”

The electric one is conditional, and the condition is charge conservation. This is checkable, and the check is worth doing because it is the kind of statement that sounds like a definition until it is tested.

Run the integration with a current in the middle of the grid, and account for the charge it moves — take it from one cell, put it in the next. Gauss’s law holds to $1.1\times10^{-14}$, exactly as it does with no source at all.

Run it again with the same current and no charge moved anywhere. The residual grows to $7.7$ and stays there.

Same equations, same grid, same current. The only difference is whether the charge the current carries exists.

What that means about charge conservation #

The usual account has charge conservation as a separate empirical fact, verified to great accuracy, which the equations happen to be compatible with.

Flux through a closed surface counts the charge inside and nothing else — that is Gauss’s law, and it is a statement of that kind at every instant separately. The question this essay asks is what keeps it true from one instant to the next. Nothing in the law itself says; it is an equation about now, with no time derivative in it, and it has to be preserved by whatever moves the fields forward.

The relation above says something stronger: Maxwell’s equations are inconsistent with a non-conserving current. Not wrong, not unphysical — inconsistent, in the sense that no field satisfies them. Feed such a current in and the system has no solution.

That is a considerable constraint on how the theory can be coupled to anything else. Any model of matter that supplies currents to Maxwell’s equations must conserve charge as an identity of its own, and this is one of the reasons the electromagnetic interaction takes the form it does in every subsequent theory.

The displacement current is where this was first forced. Ampère’s original law, without Maxwell’s term, has $\nabla\cdot\mathbf{J} = 0$ as its consistency condition, which is true only for steady currents; a capacitor being charged violates it, and it was that inconsistency rather than any experiment that produced the extra term.

The history is worth a paragraph, because the split was not obvious at the time and the modern grouping hides it. Maxwell’s own presentation had many more than four equations — twenty in the 1865 paper, in components, including ones for the potentials and for the forces on a moving charge — and it was Heaviside and Hertz who reduced them to the four vector equations everybody now writes. That reduction is what put the two constraints alongside the two evolution equations as though they were the same sort of thing.

The covariant form does the opposite, and does it more honestly. Written with the field tensor, the four become two: one equation containing the source, and one that is a pure identity — the statement that a certain combination of derivatives of an antisymmetric object vanishes. The magnetic constraint and Faraday’s law are two components of the identity, and Gauss’s law and Ampère’s law are two components of the source equation. In that arrangement the fact that one of each pair is a constraint is a matter of which index is which, and it stops looking like a coincidence.

Solving for the potentials, and the constraint that moves #

Nobody integrates the fields in practice. The potentials are the working variables, and writing the fields in terms of them changes which of the four equations are hard.

Writing $\mathbf{B}$ as the curl of something makes the constraint on $\mathbf{B}$ an identity rather than an equation, which is one of the two reasons the potentials are used. Three different vector potentials describing the same magnetic field differ by the gradient of anything at all, and every one of them has zero divergence of its curl automatically — so the constraint is not imposed, it is unimposeable, and one of the two equations disappears from the problem instead of being solved.

Writing $\mathbf{B} = \nabla\times\mathbf{A}$ makes $\nabla\cdot\mathbf{B} = 0$ automatic: the divergence of a curl is zero, so the constraint is not a condition on $\mathbf{A}$ at all. One of the four equations disappears, having been built into the variables.

Writing $\mathbf{E} = -\nabla\phi - \partial\mathbf{A}/\partial t$ does the same for Faraday’s law. What remains is two equations for two potentials, plus a gauge choice.

Gauss’s law is still there and is still a constraint, and where it sits depends on the gauge. In the Coulomb gauge it becomes Poisson’s equation for the scalar potential, solved anew at every instant — which makes the potential depend instantaneously on the charge everywhere, an unsettling feature that turns out to be gauge artefact rather than physics. In the Lorenz gauge every potential obeys a wave equation, nothing propagates instantaneously, and the constraint has been absorbed into the gauge condition — which is where the field that points where the charge is now gets its apparent instantaneity from.

This is a general pattern rather than an oddity of electromagnetism. A constraint that is preserved by the dynamics can be moved around by a change of variables, and where it ends up is a matter of convenience.

Counting what may be chosen #

There is an accounting behind all of this that makes the split between the two kinds of equation quantitative rather than a matter of presentation.

Specifying a vector field looks like specifying three numbers at every point, and the count is where the constraints show. Six numbers for $\mathbf{E}$ and $\mathbf{B}$ together, two of them not free — which is why an electromagnetic wave has two polarisations and not three, and why the count comes out the same however the field is written. The constraints are not restrictions on the solutions; they are a statement about how many solutions there are.

The two fields have six components between them, so a naive count says six numbers must be given at every point to start an integration. The two constraints remove one condition each, leaving four.

Two of those four are the wave’s two polarisations and the other two are their time derivatives, which is exactly what a wave equation for two independent quantities needs. So the count works out, and it works out only because the constraints are there.

That is the general shape of a gauge theory’s bookkeeping. The field has more components than the physics has degrees of freedom; some of the excess is removed by constraints on the initial data, and the rest by the gauge freedom, which is the statement that different field configurations can describe the same physics. Counting them correctly is how the number of polarisations of a photon comes out as two rather than four, and it is the same count that gives a gravitational wave two rather than ten.

Where the same structure turns up #

The pattern is worth naming because it recurs everywhere the field has a gauge freedom.

General relativity has the same structure and it is not an analogy. There are evolution equations for part of the metric and constraints on the rest, preserved exactly by the evolution — and a gravitational wave’s two polarisations are the surviving freedom after the constraints have taken their share, exactly as light’s two are. Numerical relativity spends much of its effort on keeping those constraints preserved against the accumulation of arithmetic error, which is the practical form of the same statement.

General relativity. Einstein’s equations split into six evolution equations and four constraints — the Hamiltonian and momentum constraints — which are preserved by the evolution as an identity of the Bianchi kind. Numerical relativity is largely the art of arranging that they stay preserved in a discrete scheme, and formulations that make the constraint violations decay rather than grow are a substantial part of the subject.

Fluid mechanics. For an incompressible flow, $\nabla\cdot\mathbf{v} = 0$ is a constraint rather than an evolution equation, and the pressure is not a thermodynamic variable at all but whatever it has to be to keep the constraint satisfied.

And any gauge theory. Gauss’s law generalises to a constraint on the physical states of the theory, and in the quantum version it becomes a statement about which states are allowed rather than about which fields are — the same kind of restriction an exclusion principle imposes, arrived at from a symmetry instead.

The common thread is that a constraint preserved by the evolution is not an extra law. It is a restriction on the initial data, and the price of having one is that some of the variables are not independent.

What a numerical scheme has to be careful about #

The clean result in the first figure is a property of how the equations were discretised as well as of the equations, and the difference is worth spelling out because it is where the subject becomes practical.

The scheme places the electric and magnetic components at different points — the field components on cell edges, their curls on cell faces — so that the discrete divergence of the discrete curl is zero term by term, by cancellation of pairs. Nothing is approximated in that cancellation, which is why the residual is at rounding rather than at the scheme’s truncation error, and why refining the grid does not change it.

Choose the placements badly — put everything at cell centres, say — and the cancellation is no longer exact. The constraint then drifts at the rate of the truncation error, which is small at first and grows, and after enough steps the field being computed does not satisfy Gauss’s law by an amount that matters. That is not a hypothetical: it is the standard failure mode of particle-in-cell plasma simulations, where the current deposited by the particles does not exactly satisfy the discrete continuity equation, and it is why such codes either use a charge-conserving deposition or periodically solve a Poisson equation to clean the divergence away.

The general moral is one that applies well beyond electromagnetism. A conserved quantity of the continuous equations is not automatically conserved by a discretisation of them, and whether it is depends on structure rather than on accuracy. A more accurate scheme that breaks the structure is often worse than a cruder one that keeps it.

There is a way of putting the whole thing that is worth carrying away. Ask what data are needed to predict the future of the electromagnetic field and the answer is: the two fields everywhere at one instant, subject to two conditions. Ask what the theory then guarantees and the answer is: the conditions will still be satisfied at every later instant, one of them unconditionally and one of them provided the sources conserve charge. Neither guarantee is a separate law; both are consequences of the same identity, applied twice.

Testing a condition the theory cannot do without #

If a non-conserving current leaves the equations with no solution, then charge conservation is not a prediction to be checked in the ordinary way — a violation would not falsify a number, it would leave the whole framework with nothing to say. That makes the experimental limits worth knowing, because they are limits on something the theory treats as structural.

The sharpest test is whether an electron can decay. It is the lightest particle carrying charge, so there is nothing charged for it to decay into; a decay to a neutrino and a photon would destroy one unit of charge outright. Detectors built for neutrino physics watch enormous volumes of material for the characteristic photon such a decay would leave, and the bound on the electron’s lifetime against that channel now stands beyond $10^{28}$ years — some eighteen orders of magnitude longer than the age of the universe, from watching a few hundred tonnes of liquid for a few years.

Other channels are bounded the same way: nuclear transitions in which a bound electron simply vanishes, and searches for charge-violating decays of heavier particles. None has ever been seen.

What makes the null results more than routine is the theoretical company they keep. A theory in which charge is not conserved cannot have the gauge symmetry that this essay’s identity comes from, and models constructed to break it in a small way turn out to predict consequences that are not small — a photon with a mass, or a violent emission of soft photons whenever a charge disappears. The experimental limits and the structural argument agree, which is the comfortable situation: the thing the equations cannot tolerate is also the thing nobody can find.

The same split, in a circuit #

The distinction between an equation that steps and an equation that constrains is not confined to fields, and the everyday instance is a circuit.

A capacitor supplies $i = C\,\mathrm{d}v/\mathrm{d}t$ and an inductor supplies $v = L\,\mathrm{d}i/\mathrm{d}t$: those are evolution equations, and given the state now they give it a moment later. A resistor supplies $v = iR$, and Kirchhoff’s two laws supply the rest — none of which contains a derivative. They are constraints, exactly like the two divergence conditions, and a circuit’s equations are therefore a mixture of the two kinds rather than a set of ordinary differential equations.

The consequences for a simulator are the ones this essay has already described in the field case. The constraints have to hold at every step, and whether they do depends on the structure of the discretisation rather than on its accuracy. The standard failure is a nonlinear capacitor: integrate $C(v)\,\mathrm{d}v/\mathrm{d}t$ step by step and the charge that ends up on the plates depends on the path taken through the timestep, so charge is quietly created or destroyed and the simulated node voltages drift. The fix used by every serious circuit simulator is to make charge the integrated variable — to model the device as $q(v)$ and differentiate that — so that the discrete scheme conserves charge by construction.

Which is the staggered grid’s argument in different clothing. In both cases the quantity that must not drift is protected by choosing variables in which its conservation is an identity, and in both cases the alternative is to compute the drift periodically and subtract it, which works and is an admission that the formulation was wrong.

Where the model runs out #

The exactness in the figure is a property of the discretisation as well as of the equations. The scheme used is a staggered one, in which the two divergences are taken on the same lattice locations the curls were, so the discrete identity $\nabla\cdot(\nabla\times) = 0$ holds term by term. On a badly chosen grid it does not, and constraint violations grow — which is a real difficulty in plasma simulation and is why divergence-cleaning schemes exist.

That the field lines of a dipole close on themselves is the picture of the constraint, and it is worth being precise about what is being claimed. The lines having no ends is $\nabla\cdot\mathbf{B} = 0$ drawn. That no process can give them ends is the statement this essay is about — not an observation that none has been seen, but a consequence of the evolution equations, which cannot produce a divergence where there was none.

The boundary is where the constraints fail, legitimately. A perfectly conducting wall carries surface charge, so $\nabla\cdot\mathbf{E}$ is not zero there, and the residuals here are measured over the interior for that reason. A constraint holding in the interior and being sourced at the boundary is the normal situation rather than an artefact.

A magnetic monopole is not simply a particle to be added. Since $\nabla\cdot\mathbf{B}$ cannot change, admitting monopoles means changing the equations: a magnetic charge density on the right of one, a magnetic current in Faraday’s law, and — most disruptively — no globally defined vector potential, since $\mathbf{B}$ is no longer a curl. Dirac’s observation that a consistent quantum treatment then forces electric charge to be quantised is the reason the possibility is still taken seriously.

And nothing here is about whether the constraints are true. They are conditions on the initial data, and the theory says only that they persist. That the universe was set up with $\nabla\cdot\mathbf{B} = 0$ everywhere is an observation, and a rather large one to have to make.

The ladder from here #

Later rungs on this anchor: the equations in potential form and what each gauge is good for; the covariant form, in which the four equations become two and the constraint structure becomes an antisymmetry; Dirac’s quantisation condition and what a single monopole anywhere would imply about every charge; and the initial-value formulation, where the question of which data may be freely chosen is answered properly.

The neighbouring ladders are the term that made light, where the consistency requirement first forces a change to the equations, counting what comes out, which is the constraint this essay watches, and the potentials that are not unique, where one of the two constraints is made to disappear by a change of variables.